Flat (2, 3, 5)-Distributions and Chazy’s Equations

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2, 3, 5)-distributions determined by a single function of the form F (q), the vanishing condition for the curvature invariant is given by a 6 order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7 order nonlinear ODE described in Dunajski and Sokolov. We show that the 6 order ODE can be reduced to a 3 order nonlinear ODE that is a generalised Chazy equation. The 7 order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2, 3, 5)distributions not of the form F (q) = q. We also give 4-dimensional split signature metrics where their twistor distributions via the An–Nurowski construction have split G2 as their group of symmetries.